Power Congruence on the 5040 Cell
Abstract
Five small multiplicative orders and coprime CRT determine the power residue on the 5040 cell.
Theorem 1.1 (The common residue is 2241).
Proof. Machine-checked in Lean as D5/S3/Arith/GoldenResource/GoldenCell5040Congruence.goldenCell5040_modEq_2241 (✓ std3). ∎
Source. Repository-derived.
Commentary.
The hypothesis is membership in the set containing 5040, 10080, 15120, 20160, 30240 and 60480. Write each member as a power of three times m. The exponent of three is two or three, and m is 560, 1120 or 2240.
The multiplicative orders of three modulo 16, 32, 64, 5 and 7 are respectively 4, 8, 16, 4 and 6. Each relevant order divides the member n, so the power of three with exponent n is congruent to one modulo each prime-power factor of m. Coprime CRT combines these congruences.
Since m divides 2240, the target 2241 has residue one modulo m. Both the target and the power of three are divisible by the three-primary factor of n. A second coprime CRT step gives the claim. The large power is kept symbolic throughout the synthesis.
This is a repository-derived statement. The upstream search reported OEIS A066601 as the general sequence of power residues, and did not find this six-member statement in the sources searched. That search was not exhaustive and establishes no claim of literature priority.
References
- Truth anchor:
D5/S3/Arith/GoldenResource/GoldenCell5040Congruence.goldenCell5040_modEq_2241